Doob decomposition theorem - Alchetron, the free social encyclopedia

In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero. The theorem was proved by and is named for Joseph L. Doob.

Statement of the theorem

Let (Ω, F, ℙ) be a probability space, I = {0, 1, 2, . . . , N} with N ∈ ℕ or I = ℕ₀ a finite or an infinite index set, (F_n)_n∈I a filtration of F, and X = (X_n)_n∈I an adapted stochastic process with E[|X_n|] < ∞ for all n ∈ I. Then there exists a martingale M = (M_n)_n∈I and an integrable predictable process A = (A_n)_n∈I starting with A₀ = 0 such that X_n = M_n + A_n for every n ∈ I. Here predictable means that A_n is F_n−1-measurable for every n ∈ I {0}. This decomposition is almost surely unique.

Corollary

A real-valued stochastic process X is a submartingale if and only if it has a Doob decomposition into a martingale M and an integrable predictable process A that is almost surely increasing. It is a supermartingale, if and only if A is almost surely decreasing.

Remark

The theorem is valid word by word also for stochastic processes X taking values in the d-dimensional Euclidean space ℝ^d or the complex vector space ℂ^d. This follows from the one-dimensional version by considering the components individually.

Existence

Using conditional expectations, define the processes A and M, for every n ∈ I, explicitly by

and

where the sums for n = 0 are empty and defined as zero. Here A adds up the expected increments of X, and M adds up the surprises, i.e., the part of every X_k that is not known one time step before. Due to these definitions, A_n+1 (if n + 1 ∈ I) and M_n are F_n-measurable because the process X is adapted, E[|A_n|] < ∞ and E[|M_n|] < ∞ because the process X is integrable, and the decomposition X_n = M_n + A_n is valid for every n ∈ I. The martingale property

E [ M n − M n − 1 | F n − 1 ] = 0 a.s.

also follows from the above definition (2), for every n ∈ I {0}.

Uniqueness

To prove uniqueness, let X = M' + A' be an additional decomposition. Then the process Y := M − M' = A' − A is a martingale, implying that

E [ Y n | F n − 1 ] = Y n − 1 a.s.,

and also predictable, implying that

E [ Y n | F n − 1 ] = Y n a.s.

for any n ∈ I {0}. Since Y₀ = A'₀ − A₀ = 0 by the convention about the starting point of the predictable processes, this implies iteratively that Y_n = 0 almost surely for all n ∈ I, hence the decomposition is almost surely unique.

Proof of the corollary

If X is a submartingale, then

E [ X k | F k − 1 ] ≥ X k − 1 a.s.

for all k ∈ I {0}, which is equivalent to saying that every term in definition (1) of A is almost surely positive, hence A is almost surely increasing. The equivalence for supermartingales is proved similarly.

Example

Let X = (X_n)_n∈ℕ₀ be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. F_n = σ(X₀, . . . , X_n) for all n ∈ ℕ₀. By (1) and (2), the Doob decomposition is given by

A n = ∑ k = 1 n ( E [ X k ] − X k − 1 ) , n ∈ N 0 ,

and

M n = X 0 + ∑ k = 1 n ( X k − E [ X k ] ) , n ∈ N 0 .

If the random variables of the original sequence X have mean zero, this simplifies to

A n = − ∑ k = 0 n − 1 X k and M n = ∑ k = 0 n X k , n ∈ N 0 ,

hence both processes are (possibly time-inhomogenious) random walks. If the sequence X = (X_n)_n∈ℕ₀ consists of symmetric random variables taking the values +1 and −1, then X is bounded, but the martingale M and the predictable process A are unbounded simple random walks (and not uniformly integrable), and Doob's optional stopping theorem might not be applicable to the martingale M unless the stopping time has a finite expectation.

Application

In mathematical finance, the Doob decomposition theorem can be used to determine the largest optimal exercise time of an American option. Let X = (X₀, X₁, . . . , X_N) denote the non-negative, discounted payoffs of an American option in a N-period financial market model, adapted to a filtration (F₀, F₁, . . . , F_N), and let ℚ denote an equivalent martingale measure. Let U = (U₀, U₁, . . . , U_N) denote the Snell envelope of X with respect to ℚ. The Snell envelope is the smallest ℚ-supermartingale dominating X and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity. Let U = M + A denote the Doob decomposition with respect to ℚ of the Snell envelope U into a martingale M = (M₀, M₁, . . . , M_N) and a decreasing predictable process A = (A₀, A₁, . . . , A_N) with A₀ = 0. Then the largest stopping time to exercise the American option in an optimal way is

τ max := { N if A N = 0 , min { n ∈ { 0 , … , N − 1 } ∣ A n + 1 < 0 } if A N < 0.

Since A is predictable, the event {τ_max = n} = {A_n = 0, A_n+1 < 0} is in F_n for every n ∈ {0, 1, . . . , N − 1}, hence τ_max is indeed a stopping time. It gives the last moment before the discounted value of the American option will drop in expectation; up to time τ_max the discounted value process U is a martingale with respect to ℚ.

Generalization

The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.

References

Doob decomposition theorem Wikipedia

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